<h2>题目编号 : 288</h2>
<div style="color:#666;font-size:80%;">17 April 2010</div><br />
<div class="problem_content">
<p>
For any prime <var>p</var> the number N(<var>p</var>,<var>q</var>) is defined by
N(<var>p</var>,<var>q</var>) = <img src='images/symbol_sum.gif' width='11' height='14' alt='&sum;' border='0' style='vertical-align:middle;' /><img src="" style="display:none;" alt="_(" /><sub><var>n</var>=0 to <var>q</var></sub><img src="" style="display:none;" alt=")" /> T<img src="" style="display:none;" alt="_(" /><sub><var>n</var></sub><img src="" style="display:none;" alt=")" />*<var>p</var><img src="" style="display:none;" alt="^(" /><sup><var>n</var></sup><img src="" style="display:none;" alt=")" /><br /> with T<img src="" style="display:none;" alt="_(" /><sub><var>n</var></sub><img src="" style="display:none;" alt=")" /> generated by the following random number generator:</p>
<p>
S<img src="" style="display:none;" alt="_(" /><sub>0</sub><img src="" style="display:none;" alt=")" /> = 290797<br />
S<img src="" style="display:none;" alt="_(" /><sub><var>n</var>+1</sub><img src="" style="display:none;" alt=")" /> = S<img src="" style="display:none;" alt="_(" /><sub><var>n</var></sub><img src="" style="display:none;" alt=")" /><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> mod 50515093<br />
T<img src="" style="display:none;" alt="_(" /><sub><var>n</var></sub><img src="" style="display:none;" alt=")" /> = S<img src="" style="display:none;" alt="_(" /><sub><var>n</var></sub><img src="" style="display:none;" alt=")" /> mod <var>p</var>
</p>
<p>
Let Nfac(<var>p</var>,<var>q</var>) be the factorial of N(<var>p</var>,<var>q</var>).<br />
Let NF(<var>p</var>,<var>q</var>) be the number of factors <var>p</var> in Nfac(<var>p</var>,<var>q</var>).
</p>
<p>
You are given that NF(3,10000) mod 3<img src="" style="display:none;" alt="^(" /><sup>20</sup><img src="" style="display:none;" alt=")" />=624955285.
</p>
<p>
Find NF(61,10<img src="" style="display:none;" alt="^(" /><sup>7</sup><img src="" style="display:none;" alt=")" />) mod 61<img src="" style="display:none;" alt="^(" /><sup>10</sup><img src="" style="display:none;" alt=")" />
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